Abstract
In this chapter, we prove the following results.
-
(1)
\(\mathsf{Z_2} +\mathsf{HP}\) is equiconsistent with \(\mathsf{ZFC}\) .
-
(2)
\(\mathsf{Z_3} +\mathsf{HP}\) is equiconsistent with \(\mathsf{ZFC} + \) “there exists a remarkable cardinal”.
-
(3)
\(\mathsf{Z_4} +\mathsf{HP}\) implies that \(0^{\sharp }\) exists.
As a corollary, “\(\mathsf{HP}\) implies that \(0^{\sharp }\) exists” is neither provable in \(\mathsf{Z_2}\) nor in \(\mathsf{Z_3}\), i.e. \(\mathsf{Z_4}\) is the minimal system of higher-order arithmetic for proving that “\(\mathsf{HP}\) implies that \(0^{\sharp }\) exists”.
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Notes
- 1.
The property \(\varSigma = 0^{\sharp }\) is \(\varPi _1\) over \((\mathsf{HC}, \in )\), and therefore a \(\varPi ^1_2\) statement.
- 2.
Especially, if \(M=V\), Definition 2.8 gives us the definition of a normal measure on cardinals.
- 3.
Examples of notions of large cardinals compatible with L are: inaccessible cardinal, reflecting cardinal, Mahlo cardinal, weakly compact, indescribable cardinal, unfoldable cardinal, subtle cardinal, ineffable cardinal, 1-iterable cardinal, remarkable cardinal, 2-iterable cardinal and \(\omega \)-Erd\(\ddot{o}\)s cardinal. For definitions of these large cardinal notions, I refer to Sect. 2.1.3 and Appendix C.
- 4.
Recall that \(\mathbb {P}\) is \(\omega \)-distributive if every function \(f: \alpha \rightarrow V\) in the generic extension with \(\alpha <\omega _1\) is in the ground model.
- 5.
I would like to thank W.Hugh Woodin and Sy Friedman for pointing out this fact to me. The proof of this fact is essentially similar as the proof of Theorem 1.25.
- 6.
For the definition of K, I refer to [22].
- 7.
The Axiom of Determinacy \((\mathsf{AD})\) states that for every \(A \subseteq \mathbb {R}\), the game \(G_{A}\) is determined.
- 8.
The answer to this question is negative if \(V=\mathsf{HOD}\). For a very easy proof of the Kunen inconsistency in the case \(V=\mathsf{HOD}\), I refer to [23, Theorem 21].
- 9.
\(\mathsf{AC}\) denotes the Axiom of Choice.
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Cheng, Y. (2019). A Minimal System. In: Incompleteness for Higher-Order Arithmetic. SpringerBriefs in Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-13-9949-7_2
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